Enhanced interference management in heterogeneous wireless networks

ABSTRACT

We show that for any given muting fraction, a more constrained version of the problem of interest can be optimally solved in an efficient manner. In addition, the obtained solution is also a near-optimal solution for the original problem (for the given muting ratio). This allows us provide an algorithm that offers a good solution to the original problem with a tractable complexity. We also derive a lower complexity greedy that offers good performance and a certain worst-case performance guarantee. Simulations over an example LTE HetNet topology reveal the superior performance of the proposed algorithms and underscore the benefits of jointly exploiting partial muting of the macro and load balancing.

RELATED APPLICATION INFORMATION

This application claims priority to provisional application No.61/930,735 filed Jan. 23, 2014, entitled “Enhanced InterferenceManagement in Heterogeneous Wireless Networks”, the contents thereof areincorporated herein by reference

BACKGROUND OF THE INVENTION

The present invention relates generally to wireless networks, and moreparticularly, to enhanced interference management in heterogeneouswireless networks.

The focus of this invention is on heterogeneous wireless networks(HetNets) that are expected to be fairly common and where thetransmission points in the HetNet will be connected to each other by anon-ideal backhaul with a relatively high latency (ranging from 50milliseconds to several dozens of milliseconds).

Over such HetNets, schemes that strive to obtain all coordinatedresource management decisions within the fine slot-level granularity(typically a millisecond) are not suitable, since coordination (whichinvolves exchange of messages and signaling over the backhaul) cannot beperformed in such a fast manner. Instead, semi-static resourcemanagement schemes, where resource management over the set oftransmission points TPs is performed at two time scales, are suitablesince they are more robust towards backhaul latency.

The present invention considers one such semi-static resource managementscheme. It focuses on a cluster of TPs that includes one high powermacro TP and several low power pico TPs. The macro TP is the dominantinterferer for all other pico TPs. The management scheme attempts tojointly exploit partial muting of the macro TP, wherein the macro TP canbe made inactive for a fraction of the frame duration, and loadbalancing (a.k.a. user association), wherein users are associated to theTPs in that cluster, such that each user is associated to any one TPover the frame duration. The fraction for which the macro is muted canbe chosen from a given finite set of fractions. This scheme requireslimited coordination among TPs in the cluster which is possible under anon-ideal backhaul. The underlying coordination is performedperiodically at a coarser frame-level granularity based on averaged (notinstantaneous) slowly varying metrics that are relevant for a periodlonger than the backhaul latency. Examples of such metrics includeestimates of average rates that the users can receive from those TPsunder different configurations etc. On the other hand, the resourcemanagement that is done at a much finer sub-frame/slot level granularityinvolves no coordination among TPs and is done independently by eachactive TP based on fast changing information, such as instantaneous rateor SINR estimates, that is received directly by that TP from the usersassociated to it.

Together, partial muting of the macro transmission point TP and loadbalancing can lead to significant benefits.

The HetNet diagram of FIG. 1 shows 3 TPs and 3 users and associatedframes. TP1 is the high power macro TP and TP2 and TP3 are the low powerpico TPs. Dashed lines indicate potential association of users to TPs.TP1 is made inactive (is muted) for a fraction of the frame duration,whereas TP2 and TP3 are active throughout the frame.

Most existing works consider either exploiting only partial muting ofthe macro TP for a given user association or exploiting only userassociation for a given partial muting. Also, the prior art has adoptedan approach which relaxes the underlying discrete variables, which candegrade performance.

Accordingly, there is a need for enhanced interference management inheterogeneous wireless networks.

BRIEF SUMMARY OF THE INVENTION

The invention is directed to a computer implemented method includingvarying association of users to any one of multiple transmission pointsin a heterogeneous wireless network for managing interference oftransmissions in the network, a muting fraction being one transmissionpoint TP being inactivated or muted for a fraction of a frame durationwhile other transmission points TPs being active throughout the frameduration, determining, at a coarse time-scale, at the start of eachframe a choice of which muting fraction to select for a macro TP andwhich users to associate with each TP so that all users are associatedto one TP, by solving an optimization problem, averaging inputs to theoptimization problem varying metrics that are relevant for a periodlonger than a backhaul latency, the varying metrics including metrics assingle user rates under muting conditions and non-muting conditions witha set of muting fractions, applying a coarse time scale approach tosolving the optimization problem, the coarse scale approach being basedon frame level TP coordination of user association and macro partialmuting; and applying a fine time scale approach to solving theoptimization problem, the fine time scale approach being based onsub-frame level per-TP user scheduling without coordination.

In a similar aspect of the invention, there is provided a non-transitorystorage medium configured with instructions for being implemented by acomputer for carrying out the method of varying association of users toany one of multiple transmission points in a heterogeneous wirelessnetwork for managing interference of transmissions in the network, amuting fraction being one transmission point TP being inactivated ormuted for a fraction of a frame duration while other transmission pointsTPs being active throughout the frame duration, determining, at a coarsetime-scale, at the start of each frame a choice of which muting fractionto select for a macro TP and which users to associate with each TP sothat all users are associated to one TP, by solving an optimizationproblem, averaging inputs to the optimization problem varying metricsthat are relevant for a period longer than a backhaul latency, thevarying metrics including metrics as single user rates under mutingconditions and non-muting conditions with a set of muting fractions,applying a coarse time scale approach to solving the optimizationproblem, the coarse scale approach being based on frame level TPcoordination of user association and macro partial muting, and applyinga fine time scale approach to solving the optimization problem, the finetime scale approach being based on sub-frame level per-TP userscheduling without coordination.

These and other advantages of the invention will be apparent to those ofordinary skill in the art by reference to the following detaileddescription and the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram of a heterogeneous network (HetNet), in accordancewith the invention.

FIG. 2 is a flow diagram of the two-time scale approach, in accordancewith the invention.

FIG. 3 is a flow diagram of the assignment-based procedure, inaccordance with the invention.

FIG. 4 is a flow diagram of a greedy procedure, in accordance with theinvention.

FIG. 5 is a diagram of an exemplary computer for carrying out theinvention.

DETAILED DESCRIPTION

The present invention is based on, that for any given muting fraction, amore constrained version of the problem of interest can be optimallysolved in an efficient manner. In addition, the obtained solution isalso a near-optimal solution for the original problem (for the givenmuting ratio). This allows providing a procedure that offers a goodsolution to the original problem with a tractable complexity. Theinventive solution also includes a lower complexity greedy procedurethat offers good performance and a certain worst-case performanceguarantee. Simulations over an example LTE HetNet topology reveal thesuperior performance of the proposed solution and underscore thebenefits of jointly exploiting partial muting of the macro and loadbalancing.

FIG. 2 is a diagram of the two-time scale approach of the presentinvention. At a coarse time-scale, at the start of each frame the choiceof which muting fraction to select for the macro TP and which users toassociate with each TP (so that all users are associated to one TP each)are determined by solving an optimization problem.

The inputs to the optimization problem are averaged (not instantaneous)slowly varying metrics that are relevant for a period longer than thebackhaul latency. At a the fine time-scale, in each slot each active TPindependently does scheduling over the set of users associated with it,without any coordination with any of the other active TPs, based on fastchanging information, such as instantaneous rate or SINR estimates, thatis received directly by that TP from the users associated to it.

We introduce the optimization problem of interest:

$\max\limits_{z \in Z}{\max\limits_{{x_{u,b} \in {\{ 0.1\}}},\gamma_{u,b},{\theta_{u,b} \in {{\lbrack{0,1}\rbrack}{\forall u}}},b}\{ {\sum\limits_{u \in U}^{\;}{\sum\limits_{b \in B}^{\;}{x_{u,b}{\ln( {{{R_{u,b}^{n\; m}( {1 - z} )}\theta_{u,b}} + {R_{u,b}^{m}z\;\gamma_{u,b}}} )}}}} \}}$$\mspace{20mu}{{{s.t.{\sum\limits_{b \in B}^{\;}x_{u,b}}} = 1},{\forall{u \in U}},\mspace{20mu}{{{{{\sum\limits_{u \in U}^{\;}\gamma_{u,b}} \leq 1}\&}{\sum\limits_{u \in U}^{\;}\theta_{u,b}}} \leq {1{\forall{b \in B}}}},}$

In this optimization problem, the inputs are the set of muting fractionsZ; the average single-user rate of each user u when associated to themacro TP; as well as the average single-user rates of each user u whenassociated to each pico TP b, under muting as well as non-muting of themacro TP (R_(u,b) ^(m), R_(u,b) ^(nm)) respectively.

The variables are the indicator variable for association of user u to TPb: x_(u,b) and the non-negative allocation fractions γ_(u,b),θ_(u,b).

The constraints are: associate each user with one TP; and the sum of therespective allocation fractions over all users should not exceed unityfor each TP.

Note that in this joint problem we are trying to jointly determine whatis the best fraction (from the given set of fractions) of the frameduration for which the macro TP should be muted, and what should be theTP each user should be associated to (user association).

This requires us to first solve a simpler problem of optimizingallocation fractions for a given feasible muting fraction and userassociation. In particular, in this simpler problem for any TP, giventhe muting fraction of the macro and the set of users associated to thatTP, we have to decide how much of the portion of the frame over whichthe macro TP is muted should be assigned to each associated user and howmuch of the portion of the frame over which the macro TP is not mutedshould be assigned to each associated user.

We completely solve this simpler problem and obtain optimal allocationfractions for the simpler problem in closed form. In the optimalsolution of this simpler problem, each TP assigns resources to at-mostone of its associated users during both muting and non-muting of themacro. The remaining users are assigned resources either during mutingor during non-muting.

The insight from this result leads us to the fact that the optimalsolution to a “more constrained” problem, in which for each mutingfraction, each user is only allowed to be served either during muting orduring non-muting and not both, is also near-optimal for the originaljoint problem.

For any given muting fraction, this “more constrained problem” is givenby:

$\max\limits_{x_{u}^{m},{x_{u}^{n\; m} \in {{\{{0,1}\}}{\forall u}}}}\{ {{\sum\limits_{u \in A}^{\;}( {{x_{u}^{n\; m}{\ln( {R_{u}^{n\; m}( {1 - z} )} )}} + {x_{u}^{m}{\ln( {R_{u}^{m}z} )}}} )} - ( {{( {\sum\limits_{k \in A}^{\;}x_{k}^{n\; m}} ){\ln( {\sum\limits_{k \in A}^{\;}x_{k}^{n\; m}} )}} + {( {\sum\limits_{k \in A}^{\;}x_{k}^{m}} ){\ln( {\sum\limits_{k \in A}^{\;}x_{k}^{m}} )}}} )} \}$  s.t.(x_(u)^(n m) + x_(u)^(m)) = 1, ∀u ∈ A,

The indicator variables x_(u,b) ^(m),x_(u,b) ^(nm) are 1 when user u isassociated to TP b during the portion of the frame where the macro TP ismuted and not muted, respectively.

This more constrained problem is a load balancing problem, which using aprevious result developed by us, can be shown to be equivalent to anassignment problem, and hence optimally solvable in an efficient manner.

Then, after optimally solving the “more constrained” problem we obtainuser association (i.e., the TP that each user is associated to). Foreach TP, we collect the set of users associated to it and then obtainthe optimal allocation fractions (using the result derived earlier forthe simpler problem). Doing so allows us to determine a good solutionfor the original joint problem under the given muting fraction. Bycomparing the solutions across all feasible muting fractions in terms ofthe system utility they respectively yield (recall that the set of allfeasible muting fractions is finite), we select the best muting fractiontogether with its corresponding user association. The resultingprocedure, referred to as the assignment based procedure, is detailedbelow in the discussion of FIG. 3.

Turning now to FIG. 3, a diagram of the assignment, initially there isan input of average single-user rates under muting and non-muting, setof feasible muting fractions 301. Then there is a selecting of a mutingfraction from the feasible set, that has not been picked before 302.

For the selected muted fraction 303:

(i) Formulate the “more-constrained” problem.

(ii) Solve the equivalent assignment problem and determine userassociation

(iii) Obtain the optimal allocation fractions for the determined userassociation

(iv) Determine the system utility

If this is the first system utility determined or if it is the largestone yet determined, then designate it as the largest utility and storethe corresponding muting fraction and user association 304.

There is a check if all muting fractions have been considered 305. Ifnot, the procedure loops back to step 302. If so, the procedure outputsthe muting fraction and corresponding user association that yields thelargest utility 306.

Next, is a lower complexity sequence in which the user association foreach muting fraction is determined in a greedy fashion. To speed up thestep of determining the best (user, TP) pair we re-consider the simplerproblem and derive the change in utility upon associating a new user toany TP.

Turning now to FIG. 2, there is shown a diagram of the greedy procedure.Initially there is an input of average single-user rates under mutingand non-muting, set of feasible muting fractions 401.

The procedure, 402, selects a muting fraction from the feasible set,that has not been picked before; Define a set containing all selected(user, TP) pairs. Set that set to be the null (empty) set.

Then the procedure, 403, selects and adds to the set, the (user, TP)pair such that: the user has not been selected before and that pairoffers the best gain in system utility among all pairs containing suchusers.

There is a check if all users have been assigned a transmission point TP404. If not, the procedure loops back to 403. If so the system utilityyielded by the set of selected (user, TP) pairs is obtained 405.

Then, if this is the first system utility determined or if it is thelargest one yet determined, then designate it as the largest utility andstore the corresponding muting fraction and set of selected pairs 406.

Then there is a check if all muting fractions have been checked 407. Ifnot, the procedure loops back to 402. If so, then there is an output ofthe muting fraction and selected set corresponding to the largestutility.

The invention may be implemented in hardware, firmware or software, or acombination of the three. Preferably the invention is implemented in acomputer program executed on a programmable computer having a processor,a data storage system, volatile and non-volatile memory and/or storageelements, at least one input device and at least one output device. Moredetails are discussed in U.S. Pat. No. 8,380,557, the content of whichis incorporated by reference.

By way of example, a block diagram of a computer to support theinvention is discussed next in FIG. 5. The computer preferably includesa processor, random access memory (RAM), a program memory (preferably awritable read-only memory (ROM) such as a flash ROM) and an input/output(I/O) controller coupled by a CPU bus. The computer may optionallyinclude a hard drive controller which is coupled to a hard disk and CPUbus. Hard disk may be used for storing application programs, such as thepresent invention, and data. Alternatively, application programs may bestored in RAM or ROM. I/O controller is coupled by means of an I/O busto an I/O interface. I/O interface receives and transmits data in analogor digital form over communication links such as a serial link, localarea network, wireless link, and parallel link. Optionally, a display, akeyboard and a pointing device (mouse) may also be connected to I/O bus.Alternatively, separate connections (separate buses) may be used for I/Ointerface, display, keyboard and pointing device. Programmableprocessing system may be preprogrammed or it may be programmed (andreprogrammed) by downloading a program from another source (e.g., afloppy disk, CD-ROM, or another computer).

Each computer program is tangibly stored in a machine-readable storagemedia or device (e.g., program memory or magnetic disk) readable by ageneral or special purpose programmable computer, for configuring andcontrolling operation of a computer when the storage media or device isread by the computer to perform the procedures described herein. Theinventive system may also be considered to be embodied in acomputer-readable storage medium, configured with a computer program,where the storage medium so configured causes a computer to operate in aspecific and predefined manner to perform the functions describedherein.

Additional Information

The transformation of traditional cellular wireless networks defined bya structured layout of cells with each cell being served by a high powermacro base-station, into dense HetNets is underway. These HetNets areformed by a multitude of transmission points (or nodes) ranging fromenhanced versions of the conventional high power macro base-station tolow power pico nodes, deployed in a highly irregular fashion [3]. Fortractable resource allocation, a HetNet is partitioned into severalcoordination units with each unit comprising of a set of transmissionpoints (TPs) that include a high power macro node along with several lowpower pico nodes, as well as a set of users that those TPs should serve.In most deployments the TPs in each unit are expected to be connected toeach other by a non-ideal backhaul with a relatively high latency andoftentimes a limited capacity. As a result, only semi-staticcoordination among TPs in the unit is viable in practice, wherein eachTP independently performs resource allocation at a fine time granularity(typically a millisecond) to serve the users associated to it, butperiodically (once in several hundred milliseconds) the set of usersassociated to each TP is altered (a.k.a. load balancing). This loadbalancing requires limited coordination among TPs in a unit which ispossible under a non-ideal backhaul, and it mitigates the undesirablescenarios of TPs becoming overloaded due to too many users beingassociated with them and users being interference limited due totransmissions from too many interfering TPs [3]. Since the main sourceof interference for a user being served by a low power pico node is thehigh power macro node, a useful feature that has been added to thefourth generation LTE cellular standard is that of partially muting themacro node [1]. Under this feature the macro node can be muted (madeinactive) for a specified fraction of the available resource, where thisfraction itself can be chosen from a limited finite set.

Simulations during the standardization process have revealed that thispartial muting feature can result in useful gains. On the other hand,the study of load balancing over HetNets has received significant recentattention, cf. [3] and the references therein. Indeed, as argued in [3]the previous approaches to load balancing (such as the maximum receivedsignal strength based association) which worked well over conventionalcellular networks can be grossly inadequate and new approaches for loadbalancing are needed [2]. Recent investigations in [4, 5] reveal thatthe gains offered by partial muting can be significantly improved whenit is used in conjunction with load balancing. Both [4, 5] adopt anapproach in which the optimal method to jointly exploit these twotechniques can be obtained by solving a purely continuous optimizationproblem. Algorithms to efficiently solve the underlying continuousoptimization problem along with several related insights are thenobtained. However, that approach itself is made possible by twoassumptions: the first one being that any muting fraction in the unitinterval can be selected and the second one being that fractional userassociation (i.e., allowing a user to be associated to multiple TPsinstead of one) is permitted. We note that neither assumption ispermitted as per the current standard specification and the solutionsobtained from either [4] or [5] must thus be rounded to ensurefeasibility. Our key contributions in this work are to formulate theproblem to determine the jointly optimal partial muting fraction andload balancing without the aforementioned assumptions and tosystematically derive an efficient algorithm whose solution isapproximately optimal.

Problem Formulation

Let U denote the set of users with cardinality |U|=K and let={1, . . . ,B} denote the set of TPs. Further, suppose that the TP with index 1εB isthe macro TP and the remaining ones are all pico TPs. For each user uεUand each TP bεB, we define R_(u,b) ^(m) to be the average rate user ucan get (per unit resource) when it alone is served by TP b and when themacro TP is muted. Similarly, we define R_(u,b) ^(nm) to be the averagerate user u can get when it alone is served by TP b and when the macroTP is not muted. These rates are averaged over fast fading but depend onthe slow fading (e.g. path loss and shadowing) realization, and includethe interference from all pico TPs other than b. Clearly, R_(u,b)^(nm)≦R_(u,b) ^(m), ∀uεU & b≧2, whereas R_(u,1) ^(nm)≧R_(u,1) ^(m)=0,∀u. For convenience in exposition we make the mild assumption thatR_(u,b) ^(nm)>0, ∀u,b. We also assume that

${\frac{R_{u,b}^{n\; m}}{R_{u,b}^{m}} \neq \frac{R_{u^{\prime},b}^{n\; m}}{R_{u^{\prime},b}^{m}}},$∀b & u≠u′ and note that these two assumptions hold true almost surelyfor all typical slow fading distributions. Next, suppose that the finiteset Z contains all possible muting fractions such that each zεZrepresents a choice for the fraction of the available unit resource forwhich the macro TP is muted. The problem of interest to us is the mixedoptimization problem given in (1).

$\begin{matrix}{{\max\limits_{z \in Z}{\max\limits_{{x_{u,b} \in {\{{0,1}\}}},\gamma_{u,b},{\theta_{u,b} \in {{\lbrack{0,1}\rbrack}{\forall u}}},b}\{ {\sum\limits_{u \in U}^{\;}{\sum\limits_{b \in B}^{\;}{x_{u,b}{\ln( {{{R_{u,b}^{n\; m}( {1 - z} )}\theta_{u,b}} + {R_{u,b}^{m}z\;\gamma_{u,b}}} )}}}} \}}}\text{}\mspace{20mu}{{{s.t.{\sum\limits_{b \in B}^{\;}x_{u,b}}} = 1},{\forall{u \in U}},\mspace{20mu}{{{{{\sum\limits_{u \in U}^{\;}\gamma_{u,b}} \leq 1}\&}{\sum\limits_{u \in U}^{\;}\theta_{u,b}}} \leq {1{\forall{b \in B}}}},}} & (1)\end{matrix}$

Note that in (1) the binary valued variable x_(u,b) is one if user u isassociated to TP b and zero otherwise, so that the first set ofconstraints in (1) ensures that each user is associated with only oneTP. Consequently, {x_(u,b)}_(uεU) yields the user set associated with TPb. The continuous variables {γ_(u,b),θ_(u,b)} are referred to here asallocation fractions and their respective sums are upper bounded byunity for each TP, as depicted in the second set of constraints.

A Simplified Setup

We begin our quest to solve (1) by first considering a simpler problemwhere we focus on any one TP, and where a muting fraction zε(0,1) isgiven and users from a set A⊂U are already associated to that TP. Wefirst suppose the TP of interest to be a pico node and for notationalconvenience we omit the subscript identifying that TP in this section.The resulting problem of interest is given by

$\begin{matrix}{\max\limits_{\gamma_{u},{\theta_{u} \in {{\lbrack{0,1}\rbrack}{\forall{u \in A_{{{\sum\limits_{u}^{\;}\gamma_{u}} \leq 1},{{\sum\limits_{u}^{\;}\theta_{u}} \leq 1}}}}}}}\{ {\sum\limits_{u \in A}^{\;}{\ln( {{{R_{u}^{n\; m}( {1 - z} )}\theta_{u}} + {R_{u}^{m}z\;\gamma_{u}}} )}} \}} & (2)\end{matrix}$

The optimization problem in (2) is that of maximizing a strictly concavefunction whose optimal solution can be completely characterized. To doso, we define a vector of effective rates as R^(eff)=[R_(u) ^(nm)(1−z),R_(u) ^(m)z]_(uεA). Furthermore, we also define some functions in (3).

$\begin{matrix}{{{{h( {R_{A}^{eff},{n + 1}} )} = {{\sum\limits_{q = 1}^{n + 1}{\ln( {R_{\pi{(q)}}^{n\; m}( {1 - z} )} )}} + {\sum\limits_{q = {n + 2}}^{A}{\ln( {R_{\pi{(q)}}^{m}z} )}} - {{A}{\ln( {A} )}} + {{A}{\ln( {1 + \alpha_{\pi{({n + 1})}}} )}} - {( {{A} - n - 1} ){\ln( \alpha_{\pi{({n + 1})}} )}}}},{{g( {R_{A}^{eff},{n + 1}} )} = {{\sum\limits_{q = 1}^{n + 1}{\ln( {R_{\pi{(q)}}^{n\; m}( {1 - z} )} )}} + {\sum\limits_{q = {n + 2}}^{A}{\ln( {R_{\pi{(q)}}^{m}z} )}} - {( {n + 1} ){\ln( {n + 1} )}} - {( {{A} - n - 1} ){\ln( {{A} - n - 1} )}}}},\mspace{20mu}{{\forall n} = 0},\ldots\mspace{14mu},{{A} - 1},\mspace{20mu}{\alpha_{u} = \frac{R_{u}^{m}z}{R_{u}^{n\; m}( {1 - z} )}},{{\forall{u \in A}};}}\mspace{20mu}{{\pi\text{:}\{ {1,\ldots\mspace{14mu},{A}} \}}->{{s.t.\mspace{14mu}\alpha_{\pi{(1)}}} < {\alpha_{\pi{(2)}}\mspace{14mu}\ldots} < {\alpha_{\pi{({A})}}.}}}} & (3)\end{matrix}$Note that in defining the permutation π(.) which is determined using theeffective rates R_(A) ^(eff) in (3), we have used the fact thatα_(u)=α_(u′) is not possible for any two distinct users u≠u′. Finally,let {circumflex over (θ)}_(u),{circumflex over (γ)}_(u), ∀u denote anoptimal solution to (2) with Ô(U) denoting the optimal objective value.

Proposition 1 There exists a unique integer {circumflex over (n)}ε{0, .. . , |A|−1} such that either

$\alpha_{\pi{({\hat{n} + 1})}} \in \lbrack {{\frac{A}{\hat{n} + 1} - 1},{\frac{A}{\hat{n}} - 1}} \rbrack$in which case the unique optimal solution to (2) is ({circumflex over(θ)}_(π(k)),{circumflex over (γ)}_(π(k)))

$= \{ \begin{matrix}{( {\frac{1 + \alpha_{\pi{({\hat{n} + 1})}}}{A},0} ),} & {{{\forall k} = 1},\ldots\mspace{14mu},\hat{n},} \\{\begin{pmatrix}{{1 - \frac{\hat{n}( {1 + \alpha_{\pi{({\hat{n} + 1})}}} )}{A}},} \\{1 - \frac{( {{A} - \hat{n} - 1} )( {1 - \alpha_{\pi{({\hat{n} + 1})}}} )}{{A}\alpha_{\pi{({\hat{n} + 1})}}}}\end{pmatrix},} & {{k = {\hat{n} + 1}},} \\{( {0,\frac{1 + \alpha_{\pi{({\hat{n} + 1})}}}{{A}\alpha_{\pi{({\hat{n} + 1})}}}} ),} & {{{\forall k} = {\hat{n} + 2}},\ldots\mspace{14mu},{A},}\end{matrix} $yielding Ô(A)=h(R_(A) ^(eff),{circumflex over (n)}+1), or

${{{\alpha_{\pi{({\hat{n} + 1})}} < {\frac{A}{\hat{n} + 1} - 1}}\&}\alpha_{\pi{({\hat{n} + 2})}}} > {\frac{A}{\hat{n} + 1} - 1}$in which case the unique optimal solution to (2) is

$\begin{matrix}{( {{\hat{\theta}}_{\pi{(k)}},{\hat{\gamma}}_{\pi{(k)}}} ) = \{ \begin{matrix}{( {\frac{1}{\hat{n} + 1},0} ),} & {{{\forall k} = 1},\ldots\mspace{14mu},{\hat{n} + 1},} \\( {0,\frac{1}{{A} - \hat{n} - 1}} ) & {{{\forall k} = {\hat{n} + 2}},\ldots\mspace{14mu},{A},}\end{matrix} } & (4)\end{matrix}$yielding Ô(A)=g(R_(A) ^(eff),{circumflex over (n)}+1).

The problem in (2) is a convex optimization problem for which the K.K.Tconditions are both necessary and sufficient. These K.K.T conditionsinclude the following:

$\begin{matrix}{{{\frac{R_{u}^{n\; m}( {1 - z} )}{{{R_{u}^{n\; m}( {1 - z} )}\theta_{u}} + {R_{u}^{m}z\;\gamma_{u}}} + \eta_{u}} = \lambda}{{{\frac{R_{u}^{m}z}{{{R_{u}^{n\; m}( {1 - z} )}\theta_{u}} + {R_{u}^{m}z\;\gamma_{u}}} + \vartheta_{u}} = \beta},}} & (5)\end{matrix}$along with the complementary slackness conditionsη_(u)θ_(u)=0,Θ_(u)γ_(u)=0, ∀u and (1−Σ_(u)θ_(u))λ=0, (1−Σ_(u)γ_(u))β=0,and the feasibility ones θ_(u),γ_(u)ε(0,1), ∀u, Σ_(u)θ_(u)≦1,Σ_(u)γ_(u)≦1. Note that λ,β,{η_(u),Θ_(u)}_(uε) are non-negativeLagrangian variables. Manipulating the first two K.K.T conditions in (5)along with the complementary slackness conditions, we see that for anytwo distinct users u, u′εθ_(u)γ_(u)≠0 & θ_(u′)γ_(u′)≠0

α_(u)=α_(u′),which is a contradiction. Consequently, there can be at-most one user ufor which θ_(u)γ_(u)≠0.From the remaining conditions, we can further deduce that

$\theta_{u} = { 0\Rightarrow\gamma_{u}  = {{{\frac{1}{\beta}\&}\frac{\beta}{\lambda}} \leq \alpha_{u}}}$$\gamma_{u} = { 0\Rightarrow\theta_{u}  = {{{\frac{1}{\lambda}\&}\frac{\beta}{\lambda}} \geq \alpha_{u}}}$

Then, using the aforementioned observations with further algebraicmanipulations we can characterize the optimal solution as claimed in theproposition, where we note that the uniqueness of the optimal solutionfollows from the strict concavity of the objective in (2). We note thatProposition 1 proved above is closely related to a result stated withoutproof in [4]. We also note that the observation that at-most one usercan be served by a pico node both during muting and non-muting has beenproved in [5]. We next introduce another useful result that will beinvoked to establish the performance guarantee of an algorithm proposedlater in the sequel. Towards that end, for the problem in (2) let usdenote any solution in which θ_(u)γ_(u)=0 ∀uε as α solution with anorthogonal split of users. Notice that in Proposition 1, the optimalsolution in (4) is a solution with an orthogonal split of users but theoptimal solution in the other case is not.

Proposition 2 The optimal among all solutions to (2) having anorthogonal split of users can be determined by solving (6).

$\begin{matrix}{{\max\limits_{x_{u}^{m},{x_{u}^{nm} \in {{\{{0,1}\}}{\forall u}}}}\{ {{\sum\limits_{u \in A}\;( {{x_{u}^{nm}{\ln( {R_{u}^{nm}( {1 - z} )} )}} + {x_{u}^{m}{\ln( {R_{u}^{m}z} )}}} )} - ( {{( {\sum\limits_{k \in A}\; x_{k}^{nm}} ){\ln( {\sum\limits_{k \in A}\; x_{k}^{nm}} )}} + {( {\sum\limits_{k \in A}\; x_{k}^{m}} ){\ln( {\sum\limits_{k \in A}\; x_{k}^{m}} )}}} )} \}}\mspace{79mu}{{{s.t.( {x_{u}^{nm} + x_{u}^{m}} )} = 1},{\forall{u \in A}},}} & (6)\end{matrix}$Moreover, the optimal solution determined from (6) yields an objectivevalue no less than Ô(A)−ln(2).

Consider any solution to (2) with an orthogonal split of users anddefine sets A^(nm)={uεA:θ_(u)>0} and A^(m)={uε:γ_(u)>0}. Note here thatwithout loss of generality we can assume that A=A^(nm)∪A^(m) and thatΣ_(uεA) _(nm) θ_(u)=1 and Σ_(uεA) _(m) γ_(u)=1. Then, using thearithmetic mean−geometric mean (AM−BM) inequality we can deduce that theobjective value yielded by the solution at hand is upper bounded byΣ_(uεα) _(nm) ln(R_(u) ^(nm)(1−z))−|A^(nm)|ln|^(nm)|+Σ_(uεA) _(m)ln(R_(u) ^(m)(1−z))−|A^(m)|ln|A^(m)|. This proves that the optimal amongall solutions with an orthogonal split of users can be determined uponsolving (6). Next, letting O′(A) denote the optimal objective value of(6), we note that in the case where the optimal solution of (2) is givenby (4), we must have that Ô(A)=Ô′(A). Further, in the remaining casegiven in Proposition 1, we see that sinceln(R _(π({circumflex over (n)}+1))^(nm)(1−z)θ_(π({circumflex over (n)}+1)) +R_(π({circumflex over (n)}+1)) ^(m) zγ_(π({circumflex over (n)}+1)))≦ln(2)+max{ln(R_(π({circumflex over (n)}+1))^(nm)(1−z)θ_(π({circumflex over (n)}+1))),ln(R_(π({circumflex over (n)}+1)) ^(m) zγ _(π({circumflex over (n)}+1)))},we can again invoke the AM−GM inequality and verify that{circumflex over (O)}(A)≦ln(2)+max{g(R _(A) ^(eff),{circumflex over(n)}+1),g(R _(A) ^(eff),{circumflex over (n)})}≦ln(2)+O′(A).which proves the proposition. Next, we will characterize the impact ofadding an additional user on the optimal solution of (2). Suppose thatthe current optimal solution is either Ô(A)=h(R_(A) ^(eff),{circumflexover (n)}+1) or Ô(A)=g(R_(A) ^(eff),{circumflex over (n)}+1) for some{circumflex over (n)}. Then, suppose that a new user u′ε\A is added tothe set A to obtain the set A′=A∪{u′}. Let R_(A′) ^(eff) denote the newvector of effective rates with {{tilde over (α)}_(u)}_(uεA′) and {tildeover (π)}:{1, . . . , |A′|}→A′ denoting the scalars and the permutationdetermined from it, respectively, following (3). We offer the followingresult whose proof follows upon using Proposition 1 along with somemanipulations and is omitted for brevity. The key message from theresult stated below is that the integer n which determines the optimalsolution either remains unchanged or is incremented by one, upon addinga new user. This result is very useful in enabling complexity reductionin a greedy algorithm introduced later in the sequel.

Proposition 3 Upon adding a new user, the optimal solution can bedetermined to be

${\hat{O}( ' )} = \{ \begin{matrix}{{h( {R_{A^{\prime}}^{eff},{\hat{n} + 1}} )},} & {{{\overset{\sim}{\alpha}}_{\overset{\sim}{\pi}{({\hat{n} + 1})}} \in \lbrack {{\frac{{A} + 1}{\hat{n} + 1} - 1},{\frac{{A} + 1}{\hat{n}} - 1}} \rbrack},} \\{{h( {R_{A^{\prime}}^{eff},{\hat{n} + 2}} )},} & {{{\overset{\sim}{\alpha}}_{\overset{\sim}{\pi}{({\hat{n} + 2})}} \in \lbrack {{\frac{{A} + 1}{\hat{n} + 2} - 1},{\frac{{A} + 1}{\hat{n} + 1} - 1}} \rbrack},} \\{{g( {R_{A^{\prime}}^{eff},{\hat{n} + 1}} )},} & {{{{{\overset{\sim}{\alpha}}_{\overset{\sim}{\pi}{({\hat{n} + 1})}} < {\frac{{A} + 1}{\hat{n} + 1} - 1}}\&}{\overset{\sim}{\alpha}}_{\overset{\sim}{\pi}{({\hat{n} + 2})}}} > {\frac{{A} + 1}{\hat{n} + 1} - 1}} \\{{g( {R_{A^{\prime}}^{eff},{\hat{n} + 2}} )},} & {Otherwise}\end{matrix} $

The following corollary completes the simplified setup by covering allthe remaining scenarios of interest that can arise when a set of usersare associated to any given TP. Corollary 1 Optimal solutions for (2) inthe two corner cases, z=0 & z=1, respectively, can be determined as

$\begin{matrix}{( {{\hat{\theta}}_{u},\gamma_{u}} ) = \{ {\begin{matrix}{( {\frac{1}{A},0} ),} & {{z = 0},} \\( {0,\frac{1}{A}} ) & {{z = 1},}\end{matrix}{\forall{u \in A}}} } & (7)\end{matrix}$Thus, in each corner case a solution with an orthogonal split of usersis also optimal with respect to (2). Further, for the macro TP and anymuting fraction, the solution with an orthogonal split of users given by

${( {{\hat{\theta}}_{u},\gamma_{u}} ) = ( {\frac{1}{A},0} )},$∀uεA is an optimal solution with respect to (2).

Notice that Ô:2→can be viewed as a set function where Ô( ) for allnon-empty subsets ⊂,≠Φ is computed using Proposition 1 and where we candefine Ô(Φ)=0. Adopting this view, we have the following conjecture.

Conjecture 1 The set function Ô(.) is a submodular set function, i.e.,for all ⊂ ⊂ & uε\ it satisfies{circumflex over (O)}(∪{u})−{circumflex over (O)}( )≧{circumflex over(O)}(∪{u})−{circumflex over (O)}( ).  (8)We note that the conjecture can be proved in the corner cases when z=0or z=1 or when the TP is the macro TP, upon invoking the fact that thefunction −xln(x) is non-increasing and concave ∀x≧1. For all other caseswe first note that the condition in (8) is equivalent to the followingone{circumflex over (O)}(∪{u})−{circumflex over (O)}( )≧{circumflex over(O)}(‘∪{u,u′})−{circumflex over (O)}(∪{u′}),  (9)for all ⊂ & u,u′ε\. We have been able to verify that (9) holds forseveral cases but a full proof eludes us.

Approximation Procedure

Let us now return to the original problem (1) and denote its optimalobjective value (referred to here as the optimal system utility) byÔ_(P). Let us expand the optimal system utility asÔ_(P)=max_(zε)Ô_(P)(z) where Ô_(P)(z) denotes the optimal objectivevalue for a given muting fraction zεZ. To obtain an approximatelyoptimal solution for (1) we will use the insights developed in Section3. Adopting the convention that 0 ln(0)=0, we formulate a purelycombinatorial problem for any given muting ratio zεZ.

$\begin{matrix}{{\max\limits_{x_{u,b}^{m},{x_{u,b}^{n\; m} \in {{\{{0,1}\}}{\forall u}}},b}\{ {{\sum\limits_{u \in U}^{\;}{\sum\limits_{b \in B}^{\;}( {{x_{u,b}^{n\; m}{\ln( {R_{u,b}^{n\; m}( {1 - z} )} )}} + {x_{u,b}^{m}{\ln( {R_{u,b}^{m}z} )}}} )}} - {\sum\limits_{b \in U}^{\;}( {{( {\sum\limits_{k \in U}^{\;}x_{k,b}^{n\; m}} ){\ln( {\sum\limits_{k \in U}^{\;}x_{k,b}^{n\; m}} )}} + {( {\sum\limits_{k \in U}^{\;}x_{k,b}^{m}} ){\ln( {\sum\limits_{k \in U}^{\;}x_{k,b}^{m}} )}}} )}} \}}{{{s.t.{\sum\limits_{b \in}^{\;}( {x_{u,b}^{n\; m} + x_{u,b}^{m}} )}} = 1},{\forall{u \in U}},}} & (10)\end{matrix}$

We offer the following result which proves that for the given zε, theoptimal solution of (10) can be efficiently obtained and it approximatesÔ_(P)(z).

Proposition 4 The optimization problem (10) is equivalent to anasymmetric assignment problem and hence is optimally solvable inpolynomial time. Further, its optimal objective value is equal toÔ_(P)(z) when z=0 or z=1 and is no less than Ô_(P)(z)−min{K,B−1} ln(2)for all other zε(0,1).

We first establish the second part of the proposition. Considering anyoptimal solution to (1) for any given muting fraction z, we expand theoptimal objective value as Ô_(P)(z)=Σ_(bε)Ô(U^((b)), b,z) where U^((b))denotes the set of users associated with TP b. We note here thatÔ(U^((b)),b,z) denotes the optimal value obtained upon solving (2) forthe associated user set U^((b)) and TP b, and we have adopted theconvention that Ô(U^((b)),b,z)=0 when U^((b))=Φ (i.e., when U^((b)) isempty). Notice that there can at-most be min{K,B−1} pico nodes withnon-empty associated user sets. Then, for each TP b for which U^((b))≠Φ,we can invoke Proposition 2 to obtain a solution that has an orthogonalsplit of users and is optimal with respect to (2) in the two cornercases z=0,z=1 or when the TP b is the macro node, and in all other casesyields an objective value no less than Ô(U^((b)),b,z)−ln(2). Collectingsuch solutions across TPs we obtain a solution that is feasible for (10)and offers an objective value that is equal to Ô_(P)(z) in the twocorner cases and is no less than Ô_(P)(z)−min{K,B−1} ln(2).

We next prove the first part of the proposition and show that theproblem in (10) can in-fact be optimally solved in polynomial time byreformulating it as an asymmetric assignment problem. To do so, wecreate K=|U| virtual TPs for each TP b under muting and non-muting,respectively, and define the rewards of associating users to thesevirtual TPs asω_(u,b,q) ^(nm)=ln(R _(u,b) ^(nm)(1−z))−q ln(q)+(q−1)ln(q−1),ω_(u,b,q) ^(m)=ln(R _(u,b) ^(m) z)−q ln(q)+(q−1)ln(q−1),for all uεU,bεB & q=1, . . . , K. Then, consider then the combinatorialproblem

$\begin{matrix}{{\max\limits_{\;^{x_{u,b,q}^{n\; m},{x_{u,b,q}^{m} \in {{\{{0,1}\}}{\forall u}}},b,q}}{\sum\limits_{u \in}^{\;}{\sum\limits_{b \in}^{\;}{\sum\limits_{q = 1}^{K}( {{x_{u,b,q}^{n\; m}\omega_{u,b,q}^{n\; m}} + {x_{u,b,q}^{m}\omega_{u,b,q}^{m}}} )}}}}{{{s.t.{\sum\limits_{u \in}^{\;}x_{u,b,q}^{n\; m}}} \leq 1},{{\sum\limits_{u \in}^{\;}x_{u,b,q}^{m}} \leq {1{\forall{b \in B}}}},{q = 1},\ldots\mspace{14mu},K}{{{\sum\limits_{b \in}^{\;}{\sum\limits_{q = 1}^{K}( {x_{u,b,q}^{n\; m} + x_{u,b,q}^{m}} )}} = {1{\forall{u \in U}}}},}} & (11)\end{matrix}$Notice that the problem in (11) is an asymmetric assignment problem.More importantly, we will show that it is equivalent to the one in (10).

Consider any TP b and its K corresponding virtual TPs under muting andnon-muting, respectively. A key observation is that the sequence {−qln(q)+(q−1)ln(q−1)}_(q=2) ^(K) is strictly decreasing in q and negativefor all q=2, . . . , K. Thus, we deduce that for any user uεU and TP bεBwe must haveω_(u,b,1) ^(nm)≧ω_(u,b,2) ^(nm)≧ . . . ω_(u,b,K) ^(nm)&ω_(u,b,1)^(m)≧ω_(u,b,2) ^(m)≧ . . . ω_(u,b,K) ^(m).

Consequently, without loss of optimality we can assume that at any givenoptimal solution of (11), a user will be associated with virtual TP q ofTP b under muting, only if other users have already been associated withvirtual TPs 1 to q−1 of TP b under muting. The same observation holdsunder non-muting as well. As a result, supposing thatU^((b))=^(Um(b))∪U^(nm(b)), where U^(m(b))∩^(Unm(b))=Φ, denotes the userset assigned to the virtual TPs of TP b and its partition into usersassigned to virtual TPs under muting and non-muting, respectively, inthat optimal solution, we see that the sum reward obtained over thesevirtual TPs is exactly equal to Σ_(uε) _(Um(b)) ω_(u,b,1)^(m)−|U^(m(b))|ln|U^(m(b))|+Σ_(uεU) _(nm(b)) ω_(u,b,1)^(nm)−|U^(nm(b))|ln|U^(nm(b))|. Hence, the overall sum reward is givenby Σ_(bεB)(Σ_(uεU) _(m(b)) ω_(u,b,1) ^(m)−|U^(m(b))|ln|U^(m(b))|+Σ_(uUε)_(nm(b)) ω_(u,b,1) ^(nm)−|U^(nm(b))|ln|U^(nm(b))|). However, this isexactly the sum reward obtained by assigning users in U^(m(b)) andU^(nm(b)) to TP b under muting and non-muting, respectively, in (10)(notice that due to the construction of (11), U=∪_(bεB)U^((b)) andU^((b))∩U^((b′))=Φ∀b≠b′). Thus we can conclude that any optimal solutionof (11) will yield a feasible solution to (10) having the same sumreward.

Similarly, now consider any optimal solution to (10) and letŨ^((b))=Ũ^(m(b))∪Ũ^(nm(b)) & Ũ^(m(b)) ∩Ũ^(nm(b))=Φ; ∀b be thecorresponding user associations. Note again that U=∪_(bεB)Ũ^((b)) andŨ^((b))∩Ũ^((b′))=Φ∀b≠b′. Then, by simply assigning users in Ũ^(m(b))(Ũ^(nm(b))) for each b one by one (the exact order is not important) tothe first |Ũ^(m(b))|(Ũ|^(nm(b))|) virtual TPs of TP b under muting(non-muting), we obtain a feasible assignment to (11) which yieldsexactly the same sum reward. Thus, any optimal solution of (10) willyield a feasible solution to (11) having the same sum reward. Combiningthese two deductions, we can deduce that (11) and (10) are equivalent.

An important consequence of Proposition 4 is that (10) can be solvedexactly in polynomial time via algorithms such as the Auction algorithm[6]. We now leverage Proposition 4 to construct the Assignment basedAlgorithm given in Table 1. In that algorithm for each muting fraction,we solve (10) to determine the set of users associated with each TP. Inparticular, for each muting fraction z, we denote U_(A) ^((b)) to be theuser set associated to TP b which is formed by all users uεU for whicheither {circumflex over (x)}_(u,b) ^(m) or {circumflex over (x)}_(u,b)^(nm) is one, where {{circumflex over (x)}_(u,b) ^(m),{circumflex over(x)}_(u,b) ^(nm)} denotes an optimal solution to (10), which in turn canbe determined by solving the equivalent problem in (11). The Assignmentbased Algorithm is an approximation algorithm for the problem in (1), asshown in the following theorem.

Theorem 1 The output of the Assignment based Algorithm yields a systemutility that satisfies Σ_(bε)Ô(_(A) ^((b)),b,{circumflex over(z)})≧Ô_(P)−min{K,B−1} ln(2).

The theorem follows upon noting that for each one of the finitely manymuting fractions in Z, the algorithm first obtains the optimal solutionfor (10) and then further improves it. In particular, the latterimprovement is obtained by considering each TP b and optimizing theallocation fractions for the users associated with it. Consequently,upon invoking Proposition 4 we can deduce that Σ_(bεB)Ô(U_(A)^((b)),b,z)≧Ô_(P)(z)−min{K,B−1} ln(2). Thus, we have that

$ {{{\max_{z \in}\{ {\sum_{b \in}{B{\hat{O}( {u_{A}^{(B)},b,z} )}}} \}} \geq \underset{\underset{{\hat{O}}_{P}}{︸}}{\max_{z \in}\{ {{\hat{O}}_{P}(z)} \}}} = {\min\{ {K,{B - 1}} \}{\ln(2)}}} \}.$

TABLE 1 Assignment Based Procedure 1. Initialize with {circumflex over(z)} = φ,{circumflex over (θ)} = −∞ . 2. For each muting fraction z ε Do3. Determine user associations {U_(A) ^((b))},b ε B by solving (10) 4.For each TP b ε Do 5. Solve (2) using the associated user set U_(A)^((b)) and Proposition 1, and let Ô(U_(A) ^((b)),b,z) be the resultingobjective value 6. End For 7. If {circumflex over (θ)} < Σ_(bε)Ô(U_(A)^((b)),b,z) Then set {circumflex over (z)} = z,{circumflex over (θ)} =Σ_(bε)Ô(U_(A) ^((b)),b,z) 8. End For 9. Output {circumflex over (z)},the corresponding user associations and allocation fractions.

We now provide another procedure to sub-optimally solve (1). Thisprocedure (given in Table 2) adopts a greedy approach and has a muchlower complexity than the Assignment based Algorithm. Also, aperformance guarantee can be established for this algorithm ifConjecture 1 can be proved.

TABLE 2 Greedy Procedure  1. Initialize with {circumflex over (z)} =φ,{circumflex over (θ)} = −∞ .  2. For each muting fraction z ε Z Do  3.Set U_(G) ^((b)) = φ, ∀ b ε B and S = U .  4. Repeat  5. For each user uε S Do  6. For each TP b ε B Do  7. Solve (2) using the associated userset U_(G) ^((b)) ∪ {u} and Proposition 3, and let Ô(U_(G) ^((b)) ∪{u},b,z) be the resulting objective value  8. End For  9. End For 10.Determine {circumflex over (b)} = argmax_(bεB)max_(uεS) {Ô(U_(G) ^((b))∪ {u},b,z) − Ô(U_(G) ^((b)),b,z)} followed by û = argmax_(uε) {Ô(_(G)^(({circumflex over (b)}))∪{u},{circumflex over (b)},z) − Ô(_(G)^(({circumflex over (b)})),{circumflex over (b)},z)} 11. Add user û toset U_(G) ^(({circumflex over (b)})) and remove it from S 12. Until S =φ 13. If {circumflex over (θ)} < Σ_(bεB)Ô(U_(G) ^((b)),b,z) Then set{circumflex over (z)} = z,{circumflex over (θ)} = Σ_(bεB)Ô(U_(G)^((b)),b,z) 14. End For 15. Output {circumflex over (z)} , thecorresponding user associations and allocation fractions.

Simulation Results

We now present our initial simulation results obtained for an LTE HetNetdeployment. We consider a coordination unit formed by one sector whichcontains a set of B=11 TPs formed by one macro base-station and tenlower power (pico) nodes along with K=32 users. All TPs have onetransmit antenna and all users have one receive antenna each. Each macrobase-station transmits with a power of 46 dBm whereas the transmit powerat each pico node is 35 dBm. A noise power of −104 dBm was assumed. Theother major parameters are all as per 3GPP evaluation guidelines. Forsimplicity we assume that only one muting fraction={0} is allowed andexamine the performance of the Assignment based algorithm (which we knowfrom Theorem 1 and Proposition 4 to be optimal in this case) and thegreedy algorithm. To benchmark the performance of these two algorithmswe determine the utility value for (1) as well as the average and the5-percentile spectral efficiency (SE) yielded by a baseline scheme inwhich each user independently associates to the TP from which it canobtain the highest average rate. This association scheme is alsoreferred to as the maximum SINR association [3]. Next, we determine theutility value and the average and 5-percentile SE values yielded by thegreedy algorithm, as well as the ones yielded by the assignment basedalgorithm. These values are provided in Table 3 as relative percentagegains over the respective baseline counterparts. From the results inTable 3, we see that the greedy algorithm is almost optimal and thatboth the algorithms offer significant 5-percentile SE gains withoutdegrading the average SE.

TABLE 3 Relative percentage gains Average 5-percentile Algorithm Utilitygain SE gain SE gain Greedy 17.51% 0.9% 85% Assignment based 17.57% 1.3%86%

REFERENCES

[1] 3GPP, “Evolved Universal Terrestrial Radio Access (E-UTRA) andEvolved Universal Terrestrial Radio Access Network (E-UTRAN),” in(Release 10), TS 36.300 V10.8.0, June 2012.

[2] Q. Ye, et. al., “User association for load balancing inheterogeneous cellular networks,” in IEEE Trans. on Wireless Commun.,March 2013.

[3] J. Andrews, et. al., “An overview of load balancing in HetNets: Oldmyths and open problems,” in IEEE commun. mag. (submitted), July 2013

[4] A. Bedekar and R. Agrawal, “Optimal muting and load balancing foreICIC,” Proc. IEEE WiOPT, May 2013.

[5] Q. Ye, et. al., “On/Off Macrocells and Load Balancing inHeterogeneous Cellular Networks”, in IEEE Globecom., December 2013

[6] D. Bertsekas and D. Castanon, “A forward/reverse auction algorithmfor asymmetric assignments problems,” Comp. Optimization and Appl.,1992.

From the foregoing, it can be appreciated that the present inventionachieves considerably higher system utility than existing approaches.Both the average and the 5-percentile spectral efficiencies aresignificantly improved.

The foregoing is to be understood as being in every respect illustrativeand exemplary, but not restrictive, and the scope of the inventiondisclosed herein is not to be determined from the Detailed Description,but rather from the claims as interpreted according to the full breadthpermitted by the patent laws. It is to be understood that theembodiments shown and described herein are only illustrative of theprinciples of the present invention and that those skilled in the artmay implement various modifications without departing from the scope andspirit of the invention. Those skilled in the art could implementvarious other feature combinations without departing from the scope andspirit of the invention.

The invention claimed is:
 1. A computer implemented method comprising:varying association of users to any one of multiple transmission pointsin a heterogeneous wireless network for managing interference oftransmissions in the network, a muting fraction being one transmissionpoint TP being inactivated or muted for a fraction of a frame durationwhile other transmission points TPs being active throughout the frameduration; determining, at a coarse time-scale, at the start of eachframe a choice of which muting fraction to select for a macro TP andwhich users to associate with each TP so that all users are associatedto one TP, by solving an optimization problem; averaging inputs to theoptimization problem varying metrics that are relevant for a periodlonger than a backhaul latency, the varying metrics including metrics assingle user rates under muting conditions and non-muting conditions witha set of muting fractions; applying a coarse scale approach to solvingthe optimization problem, the coarse scale approach being based on framelevel TP coordination of user association and macro partial muting; andapplying a fine scale approach to solving the optimization problem, thefine scale approach being based on sub-frame level per-TP userscheduling without coordination.
 2. The computer implemented method ofclaim 1, wherein the fine time scale approach comprises that in eachslot each active TP independently does scheduling over a set of usersassociated with it, without any coordination with any of the otheractive TPs, based on fast changing information, such as instantaneousrate or SINR estimates, that is received directly by that TP from theusers associated to it.
 3. The computer implemented method of claim 1,wherein the coarse time scale approach comprises selecting a mutingfraction from a feasible set of transmission points and users that havenot been picked before and determining system utility that includes userassociations and an optimal allocation of a muting fraction fordetermined user associations.
 4. The computer implemented method ofclaim 3, further comprising if this is the first system utilitydetermined or if it is the largest one yet determined, then designatingit as a largest utility and storing a corresponding muting fraction anduser association.
 5. The computer implemented method of claim 4, furthercomprising considering if all muting fractions have been considered indetermining the system utility and if so outputting the muting fractionand corresponding user association that yields the largest utility ofthe network.
 6. The computer implemented method of claim 2, wherein thefine time scale approach comprises selecting a muting fraction from afeasible set of muting fractions, that has not been picked before,defining a muting fraction set containing all selected user and TPpairs, and setting that muting fraction set to be a null or empty set.7. The computer implemented method of claim 6, wherein the fine timescale approach comprises selecting and adding to the muting fractionset, the user and TP pair such that the user has not been selectedbefore and that pair that offers the best gain in system utility amongall pairs containing such users.
 8. The computer implemented method ofclaim 7, further comprising considering if all users have been assigneda TP and, if so, obtaining a system utility yielded by the mutingfraction set of selected user and TP pairs.
 9. A non-transitory storagemedium configured with instructions for being implemented by a computerfor carrying out the method comprising: varying association of users toany one of multiple transmission points in a heterogeneous wirelessnetwork for managing interference of transmissions in the network, amuting fraction being one transmission point TP being inactivated ormuted for a fraction of a frame duration while other transmission pointsTPs being active throughout the frame duration; determining, at a coarsetime-scale, at the start of each frame a choice of which muting fractionto select for a macro TP and which users to associate with each TP sothat all users are associated to one TP, by solving an optimizationproblem; averaging inputs to the optimization problem varying metricsthat are relevant for a period longer than a backhaul latency, thevarying metrics including metrics as single user rates under mutingconditions and non-muting conditions with a set of muting fractions;applying a coarse scale approach to solving the optimization problem,the coarse scale approach being based on frame level TP coordination ofuser association and macro partial muting; and applying a fine scaleapproach to solving the optimization problem, the fine scale approachbeing based on sub-frame level per-TP user scheduling withoutcoordination.
 10. The non-transitory storage medium of claim 9, whereinthe fine time scale approach comprises that in each slot each active TPindependently does scheduling over a set of users associated with it,without any coordination with any of the other active TPs, based on fastchanging information, such as instantaneous rate or SINR estimates, thatis received directly by that TP from the users associated to it.
 11. Thenon-transitory storage medium of claim 9, wherein the coarse time scaleapproach comprises selecting a muting fraction from a feasible set oftransmission points and users that have not been picked before anddetermining system utility that includes user associations and anoptimal allocation of a muting fraction for determined userassociations.
 12. The non-transitory storage medium of claim 3, furthercomprising if this is the first system utility determined or if it isthe largest one yet determined, then designating it as a largest utilityand storing a corresponding muting fraction and user association. 13.The non-transitory storage medium of claim 2, further comprisingconsidering if all muting fractions have been considered in determiningthe system utility and if so outputting the muting fraction andcorresponding user association that yields the largest utility of thenetwork.
 14. The non-transitory storage medium of claim 10, wherein thefine time scale approach comprises selecting a muting fraction from afeasible set of muting fractions, that has not been picked before,defining a muting fraction set containing all selected user and TPpairs, and setting that muting fraction set to be a null or empty set.15. The non-transitory storage medium of claim 14, wherein the fine timescale approach comprises selecting and adding to the muting fractionset, the user and TP pair such that the user has not been selectedbefore and that pair that offers the best gain in system utility amongall pairs containing such users.
 16. The non-transitory storage mediumof claim 15, further comprising considering if all users have beenassigned a TP and, if so, obtaining a system utility yielded by themuting fraction set of selected user and TP pairs.